Let \(\mathbb {M}_n(\mathbb {M}_k)\)  be the set of \(n\times n\)  block matrices
with each block being a \(k\times k\)  matrix.
The element of \(\mathbb {M}_n(\mathbb {M}_k)\)  is usually written as
the bold letter \({\mathbf {A}}=[A_{ij}]_{i,j=1}^n\) , where \(A_{ij}\in \mathbb {M}_k\)  for all \(1\le i,j\le n\) .
Given \({\mathbf {A}}=[A_{ij}],{\mathbf {B}}=[B_{ij}]\in \mathbb {M}_n(\mathbb {M}_k)\) ,
the block Hadamard product of \({\mathbf {A}}\)  and \({\mathbf {B}}\)  is given as \({\mathbf {A}}\,\Box \,{\mathbf {B}}:=[A_{ij}B_{ij}]_{i,j=1}^n\) ,
where \(A_{ij}B_{ij}\)  denotes the usual matrix product of \(A_{ij}\)  and \(B_{ij}\) .
Clearly, when \(k=1\) , that is, \(A\)  and \(B\)  are \(n\times n\)  matrices with complex entries,
then the block Hadamard product coincides with the classical Hadamard product;
when \(n=1\) , it is identical with the usual matrix product.
Positive definite block matrices are most appealing and extensively studied
over the recent years since it leads to a number of versatile and elegant matrix inequalities;
see, e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}.
